In the South China Sea (SCS), 14 nonlinear internal waves are detected as they transit a synchronous array of 10 moorings spanning the waves’ generation site at Luzon Strait, through the deep basin, and onto the upper continental slope 560 km to the west. Their arrival time, speed, width, energy, amplitude, and number of trailing waves are monitored. Waves occur twice daily in a particular pattern where larger, narrower “A” waves alternate with wider, smaller “B” waves. Waves begin as broad internal tides close to Luzon Strait’s two ridges, steepening to O(3–10 km) wide in the deep basin and O(200–300 m) on the upper slope. Nearly all waves eventually develop wave trains, with larger–steeper waves developing them earlier and in greater numbers. The B waves in the deep basin begin at a mean speed of ≈5% greater than the linear mode-1 phase speed for semidiurnal internal waves (computed using climatological and in situ stratification). The A waves travel ≈5%–10% faster than B waves until they reach the continental slope, presumably because of their greater amplitude. On the upper continental slope, all waves speed up relative to linear values, but B waves now travel 8%–12% faster than A waves, in spite of being smaller. Solutions of the Taylor–Goldstein equation with observed currents demonstrate that the B waves’ faster speed is a result of modulation of the background currents by an energetic diurnal internal tide on the upper slope. Attempts to ascertain the phase of the barotropic tide at which the waves were generated yielded inconsistent results, possibly partly because of contamination at the easternmost mooring by eastward signals generated at Luzon Strait’s western ridge. These results present a coherent picture of the transbasin evolution of the waves but underscore the need to better understand their generation, the nature of their nonlinearity, and propagation through a time-variable background flow, which includes the internal tides.
The strongest nonlinear internal waves (NLIW) in the world’s oceans occur in the South China Sea (SCS; Fig. 1), where their horizontal and vertical velocities can exceed 2 and 0.7 m s−1, respectively (Klymak et al. 2006). These flows and the associated downward displacements of >200 m are great enough to hamper surface and submarine navigation. In addition, their fluid speeds can exceed their wave celerities, leading to trapped cores and elevated dissipation (R.-C. Lien et al. 2010, unpublished manuscript). The associated mixing (St. Laurent 2008) and/or transport of nutrients and prey cause pilot whales to forage preferentially in their wakes (Moore and Lien 2007).
In spite of their strength and obvious scientific and societal importance, much remains unknown regarding the waves’ dynamics and generation. They are always directed westward and usually arrive twice a day in a quasi-predictable pattern, making it clear that they originate via interactions with the barotropic tide and the two-ridge system at Luzon Strait. Because strongly nonlinear waves are not seen in satellite imagery east of about 120.5°E (Zhao et al. 2004), the waves do not appear to be generated as lee waves directly at the sill. Instead, the most attractive theory for their formation is 1) generation of a westbound internal tide via flow of the barotropic tide over the ridges followed by 2) formation of narrow nonlinear waves via instability or steepening of the internal tide (Lien et al. 2005). Still, in spite of ongoing numerical modeling efforts, the geometry and degree of nonlinearity of the internal tide generation is poorly known, as is the mechanism by which they break down into NLIW. Factors hampering progress include the large and complicated two-ridge system, the potential for three-dimensional effects such as interference between multiple sources along the ridge, the strong lateral stratification gradient and shear associated with the Kuroshio, and a relative lack of data near the generation region.
In addition, rotation fundamentally changes the dynamics of nonlinear wave formation and evolution. Unlike NLIW observed at lower latitude (Apel et al. 1985), rotation cannot be neglected in the SCS. With regard to formation, rotational dispersion can prevent or slow the breakdown into NLIW of the internal tide (Gerkema et al. 2006; Helfrich and Grimshaw 2008; Farmer et al. 2009). In addition, the internal tides themselves are dispersive, causing the semidiurnal and diurnal constituents to travel at different speeds (Zhao and Alford 2006, hereafter ZA06). Finally, the rotating analog of the Kortevrieg–de Vries (KDV) equation (which describes weakly nonlinear solitary waves in a nonrotating frame) has not been solved analytically, but it appears to not have solitary solutions (Leonov 1981).
Instead, several families of solutions have been determined numerically (Grimshaw et al. 1998) but have not been adequately explored or verified observationally. In all of these solutions, the nonlinear waves and internal tides appear to interact strongly. Hence, with rotation we lack simple expressions from KdV theory such as that for the phase speed,
where η is the wave amplitude, ce is the linear irrotational phase speed (appendix A), and α is a function of the stratification.
In contrast to KdV waves, whose speed exceeds the linear value by a prescribed amount via (1), rotational nonlinear solutions can travel either faster or slower than their linear counterparts (Helfrich and Grimshaw 2008). Accurate measurements of wave speed and comparison to linear values may therefore constrain ongoing efforts to numerically model the waves.
In this paper, we document the speed and structure of 14 NLIW as they transit a 10-element transbasin array (Fig. 1), with the aim of providing observational constraints to assist in better determination of their generation and subsequent dynamics. We have three primary goals:
(i) Use of the waves’ arrival times to infer the tidal phase at which they are generated: ZA06 noted a striking similarity between the temporal pattern of arrivals on the upper continental slope and periods of maximum westward flow, with a lag consistent with travel at ce. Our observations west of 119°E are consistent with this scenario; however, our observations in the deep basin are not, suggesting instead generation near the following eastward current and propagation at ≈(1.1–1.2)ce. The inconsistency, which may be partly due to contamination of the signals at mooring A1 (Fig. 1) by eastward signals generated at the western ridge, highlights the need for a better understanding of the waves’ generation.
(ii) Measurement of wave speed and comparison to linear values: Observed speeds always exceed ce, apparently downplaying the importance of nonhydrostatic effects (which would tend to decrease speeds below ce) and other slower-traveling solutions. In addition, larger waves travel faster than smaller waves in the deep basin, as expected for KdV waves without rotation. However, the opposite is true on the upper slope, which we attribute to modulation by the baroclinic currents of a diurnal internal tide there.
(iii) Documentation of the waves’ structure as they transit the SCS: We find that the waves begin quite broad in the deep basin, sharpening and developing wave trains as they move westward.
After first discussing the moored dataset and our methods, the results will be presented in three parts. First, the wave arrivals will be presented, and their implied relationship with the barotropic tide will be discussed. Second, our speed estimates will be presented and compared with linear values. We will explain the manner by which the internal tide can cause the smaller waves on the upper slope to travel faster than the larger waves. Third, we will document the structure of the waves following their travel onto the upper slope. Conclusions and a discussion follow.
2. Data and methods
a. Mooring descriptions
During 2006–07, 10 moorings were deployed in the SCS as part of the Nonlinear Internal Waves Initiative (NLIWI) by a combination of U.S. and Taiwanese investigators (Fig. 1; Table 1). All moorings were operational during the period 26 April–7 May 2007, which encompasses the arrivals of a set of 14 waves generated at a spring tide. Because our focus is tracking the waves across the entire array, only the overlap period is considered here.
Design and instrumentation differs from mooring to mooring, but in general the moorings are of three types, indicated in Table 1:
(i) Three pressure inverted echo sounders (PIES; A1–A3) deployed in the deep basin measured the round-trip travel time of acoustic pulses transmitted from the bottom every minute: As shown in Li et al. (2009), travel time is proportional to the upward mode-1 displacement of the thermocline. With stratification information, travel time can be inverted to yield the amplitude of the displacement; here, the raw travel time signals are sufficient to determine the arrival time and width of each wave.
(ii) Acoustic Doppler current profiler (ADCP) moorings: At N2, three 300-kHz Teledyne RDI (TRDI) Workhorse ADCP resolved currents in segments spanning about 50–545 m (approximately the upper half). At the other sites, 75-kHz TRDI Long Ranger ADCPs were used. At MP1, the instrument was mounted in a cage at 80 m looking downward. Using 16-m bins and narrowband pulses, the instrument reached 864-m depth before signals became too weak. At N1 and LR1–2–3, ADCPs were mounted several hundred meters below the surface looking upward. At these sites, the upper 10% of the sampling range is discarded because of contamination by the surface reflection of the first sidelobe. The usable range and sampling interval for each mooring is given in Table 1.
(iii) Profiling moorings: Moorings MP1 and MP2 were equipped with McLane moored profilers (MP), which crawled up and down the mooring wires at 0.25 m s−1. Each carried a Falmouth Scientific (FSI) two-axis current meter and an FSI CTD. At MP1, the profile interval of 1.5 h resolves the internal tide but was too coarse to resolve the nonlinear waves. Hence, ADCP data are used above 860 m, and MP data are used below. At MP2, only MP data are available, because its ADCP’s memory card failed. Waves were marginally resolved in the MP data, with profiles each 20 min.
Horizontal and vertical velocity (u, υ, and w) were computed at each ADCP in standard fashion by transforming raw beam-coordinate velocities to earth coordinates using accelerometers and compasses inside each instrument. All compasses were calibrated prior to deployment, and clock drifts were calculated and corrected for. Beam separation increases to several hundred meters at great range, which introduces some errors in velocity estimates when wave widths are comparable or smaller (Scotti et al. 2005). These uncertainties collectively introduce speed errors of several percent, as discussed in appendix B.
At MP1 and N2, strong currents caused appreciable layovers (as great as 56 and 220 m, respectively), causing the depth coverage to vary somewhat. To account for the resulting variable measurement depths, data were interpolated onto common pressure surfaces using pressure measured on or near each instrument. Vertical velocities at MP1, used in computing energy, were corrected for the vertical motion of the instrument by subtracting off the time derivative of the pressure signal. Associated errors are negligible because of the dominance of horizontal velocities in the energy calculation.
c. Arrival time and speed
Wave arrival times tw are determined based on the time of the greatest downward displacement (PIES) and maximum shallow westward flow (for the ADCP–MP moorings), as shown in time series showing three sample waves (7–9, Fig. 2a). Data from each mooring are plotted, with subsequent panels advanced such that wave 8 appears at the center in each. Nearly every wave was clearly identifiable in all moorings, as for these three waves. At A1, where waves are quite broad, signals are smoothed over 3 h prior to determining the arrival time. Arrivals are not calculated at N2, because signals are extremely complex given the mooring’s proximity to the sill.
Ramp et al. (2004, hereafter R04) noted alternating arrivals of stronger, narrower “A” waves, which occur at about the same time each day, with weaker, broader “B” waves occurring about an hour later each day. We observe a similar pattern, evident in Fig. 2a, where waves 7 and 9 are larger and narrower than the central wave 8. We will follow R04’s nomenclature and refer to odd waves in our study as A waves and even waves as B waves.
Because the waves on the upper slope are too sharp to be clearly seen in Fig. 2a, a fourth wave is examined in close up (Fig. 2b). Time series have been transformed to spatial coordinates by multiplication by the wave speed (determined immediately below), which decreases moving westward. As a result, a 1-h interval (indicated with thick bars at the top of each panel) spans less and less distance moving westward. At most locations, uncertainty in arrivals is <5 min because of the high signal–noise ratio. At A1, uncertainty is ≈1 h because of the broad signals. At MP1, uncertainty is ≈20 min because of the slower sample interval.
Prior to calculation of speed, a propagation track must be assumed. A generation site atop the eastern ridge of Luzon Strait near the Batan Islands (Fig. 1, G) was first selected based on past observational (R04) and numerical model results (Niwa and Hibiya 2003; Jan et al. 2008). A track was then constructed that was consistent with observed wave propagation paths from moorings and satellite imagery (selected crests shown in Fig. 1, white). The along-track distance of each mooring is then determined by assuming cylindrical spreading of wave crests as they move westward (Fig. 1, white dots). Sensitivity of the results to track choice and projection method is discussed in appendix B.
Once the track and projection method are selected, observed phase speed cpobs is simply computed for each wave between each pair of moorings by dividing the along-track distance between the moorings (Fig. 5a) by the arrival time difference.
For comparison, linear phase speeds are also computed in standard fashion for hydrostatic mode-1 waves without rotation ce, as well as for semidiurnal (cpM2) and diurnal (cpK1) internal waves in a rotating frame, as described in appendix A. Briefly, ce is first computed, following ZA06, at each location in the domain by solving a Sturm–Liouville equation using climatological stratification (validated with observations) and high-resolution bathymetry; cpM2 (1.07ce in the SCS) and cpK1 (1.5ce in the SCS) are then computed from the internal wave dispersion relations.
d. Calculation of amplitude, energy, and width
Once the wave’s speed is known, it is straightforward to transform into spatial coordinates, x = cpobs(t − tw) (Fig. 2b). Wave amplitude ηo(z) and width Δ are then computed at each depth by first integrating w in time to obtain displacement and then fitting sech2 profiles,
Though rotating solutions are not necessarily expected to have the sech2 form predicted by KdV, sample fits (black lines) demonstrate its validity for the purpose of estimating amplitude and width.
For the ADCP and MP moorings, mode-1 amplitude is then determined by fitting ηo(z) to the mode-1 eigenfunction computed using climatological stratification at each mooring site (vertically migrating scatterers contaminate some vertical velocity estimates near local dawn and dusk; we discard these). For the PIES moorings, width is easily determined, but amplitude is not computed to avoid uncertainties in converting travel time to displacement. Fits are not attempted at MP2 because of marginal temporal resolution.
At the ADCP and MP moorings, kinetic energy is calculated for each wave by integration over a period including the wave,
and subtracting off the “background” energy,
where ρo is the depth-mean density determined from climatology; H is the water depth; x− and x+ are integration limits corresponding to just before and after the wave (±30 min); and u−, υ−, and w− are velocity at x−. Energy values are most precise at LR1–LR3, where wave energy greatly exceeds the background and depth coverage is greatest. At N1 and MP1, which do not cover the full water column, the calculation is extrapolated to the bottom assuming a mode-1 profile. Values at N1 and MP1 are somewhat sensitive to the integration limits x− and x+, reflecting the increased difficulty in separating wave from internal tide energy in the deep basin.
3. Wave arrivals
a. Time series
The group of waves is next examined as a whole by plotting time series of zonal velocity at four moorings (Fig. 3 for N1 and MP1 and Fig. 4 for LR1 and LR3). For each, east–west velocity is plotted in eight successive 24-h periods. The start time for each mooring is lagged moving westward by the travel time of the waves so that they appear at approximately the same location in each plot. The arrival time of each wave is indicated with vertical dashed lines and numbers.
In the deep basin (N1; Fig. 3a), waves are evident as broad westward pulses superimposed on a quasi-sinusoidal, dominantly semidiurnal background flow. Measurements do not extend deep enough to resolve the expected deep eastward flow for mode-1 signals. At MP1, signals have sharpened noticeably, with a portion of the deep eastward flow for each wave now seen. Trailing waves begin to develop at MP1, increasing in numbers moving westward. The full depth structure is seen at the LR moorings, which measure nearly the entire water column. All waves detected show a dominant mode-1 structure. Vertical velocity at all sites (not shown) is downward at the leading edge of the wave, as expected for westward-propagating thermocline depressions.
The particular timing of A and B wave arrivals reported by R04 is evident. Specifically, two waves are seen in each 24-h period, with a stronger, narrower A wave (odd numbers) occurring nearly the same time each day (hours 15–17) and a broader, weaker B wave (even numbers) whose arrival is about an hour later each day. The pattern is similar at all moorings, in spite of some distortion moving westward because of the different travel time of each wave.
b. Relationship to barotropic tide forcing at Luzon Strait
All detected wave arrivals across the basin are summarized in Fig. 5a. With a few exceptions, all waves are detected at all moorings, beginning as broad depressions at A1 and steepening as they move westward. The waves plotted previously are circled. Waves are seen transiting each mooring, slowing toward the west as they shoal onto the upper continental slope. These arrivals will be examined in this section with the goal of inferring the phase of the barotropic tide at Luzon Strait leading to the waves’ generation. Rather inconsistent results cause us to fall short of this goal, highlighting the complicated nature of the signals near the generation region.
Our analysis requires a reliable estimate of barotropic flow past the eastern ridge. Because of a lack of data, past work employed predictions from the TPXO6.2 Ocean Topography Experiment (TOPEX)/POSEIDON global ocean tidal model evaluated near the ridge (Egbert and Erofeeva 2002). Here, these predictions are compared to observations of zonal velocity from mooring N2 (Fig. 5b), about 10 km west of the ridge top in 930 m of water. As noted above, strong currents led to significant blowdowns resulting in shallow gaps, particularly later in the record. In addition, failure of the lower ADCP resulted in no data below 530 m. The observed currents show strong downward phase propagation consistent with an upward-radiating internal tide presumably generated at the eastern ridge. Given the strength of the baroclinic currents and the incomplete depth coverage, uncertainty remains regarding the degree to which the depth average (Fig. 5c, black) represents the barotropic flow. In spite of this, the amplitude and phase of the depth-averaged currents and predictions from TPXO6.2 evaluated at G (gray) are similar. With this caveat in mind, the depth-averaged zonal currents at N2 will be used as an imperfect proxy for the barotropic tidal forcing at the sill.
Past work (ZA06) noted that the pattern of arrivals at mooring S7 on the upper continental slope (Fig. 1, gray) closely matches that of maximum westward flow at Luzon Strait, when lagged by a time consistent with travel at ce. Using satellite imagery of wave crests, Jackson (2009, hereafter J09) derived an empirical relationship between wave phase speed and bottom depth and also found consistency with generation at westward maxima. J09’s waves travel approximately 10%–15% faster than ce, as found here. However, J09’s waves are spawned at 120°E, 200 km west of the eastern ridge. As a result of these two factors, J09 infers generation at the westward maximum one day later than ZA06.
Our data west of 119°E are consistent with the results of both J09 and ZA06 (because of the periodicity, it is difficult to distinguish between the two westward peaks identified by ZA06 and J09). To demonstrate, trajectories (gray) are initiated at the eastern ridge crest each time of maximum westward flow (plotted at right, Fig. 5c) and integrated westward across the basin with a speed ce. As noted, arrivals at MP1 follow a particular pattern wherein A waves follow B waves more closely than vice versa. As seen by close agreement between arrivals at MP1 and the gray lines, both this long–short interval pattern and the absolute arrival times are well predicted by the ZA06 trajectories.
Trajectories predicted by J09 were also examined (not shown for clarity). These begin at 120°E, one day later than the ZA06 trajectories, and propagate westward at a shallower angle, also lining up well with the arrivals at MP1. West of there, J09 predictions are better than those of ZA06 because of the waves’ faster observed speeds.
However, arrival times in the deep basin, particularly at A1, are inconsistent with these scenarios, occurring about 6–8 h after the ZA06 predictions and 16–18 h before the J09 predictions. In addition, the observed trajectories are clearly shallower than the gray lines, indicating travel faster than ce. Because of the shorter implied travel times for the same arrivals, our results therefore imply generation shortly after the subsequent eastward currents. To estimate generation times, trajectories connecting the observed arrivals (Fig. 5a, dashed lines) are extrapolated from A1 back to the generation site assuming travel at cpM2. Because the mode-1 wavelength is greater than the scale over which depth changes near the sill, an alternate estimate is obtained by assuming the waves travel at the deep-water speed to A1 (Fig. 5a, straight dashed lines). The two methods yield travel times to A1 of 8.9 and 5.9 h, respectively. The intersection times (Figs. 5b,c, horizontal dashed lines) align slightly after locally maximum eastward currents, in agreement with numerical models (Buijsman et al. 2010; H. Simmons 2008, unpublished manuscript; O. Fringer 2008, personal communication).
The alternating pattern of long/short arrival intervals at MP1 is oddly similar to the observed asymmetry in the timing of the westward current beats (gray curves). If all waves travel at the same speed, this pattern would be mimicked in the wave arrival patterns. By contrast, the eastward currents do not show a similar asymmetric pattern but are more evenly spaced, as the A1 arrivals are as well.
If the arrivals at A1 are representative of westward mode-1 waves generated at the eastern ridge, these observations would suggest that the observed asymmetry develops because A waves travel faster in the deep basin (as evidenced by the shallower trajectories in Fig. 5a), which reduces the interval between them and the preceding B wave, as observed. However, preliminary modeling results (not shown) indicate that the data at A1, which lies between the two ridges, may be influenced by eastward signals generated at the western ridge, which would complicate these interpretations. In addition, the expected beam-like structure could complicate interpretation of the PIES data, which are only sensitive to the depth integral of vertical motions. The inconsistency between the generation phase implied by the western and eastern moorings highlights the difficulty in associating the regular remote arrivals with the complicated near-field structure. More study is required to ascertain the waves’ generation physics and phase.
4. Wave speed
a. Comparison of observed and linear speeds
Observed speeds are examined next and compared to linear speeds (Fig. 5d). Linear speeds for zero rotation (ce; black), semidiurnal (cpM2; gray) and diurnal internal waves (cpK1; dashed) decrease by a factor of 3 (from ∼3 to ∼1 m s−1) transiting the array, primarily because of the decrease in water depth (appendix A). Observed speeds are never less than cpM2, but they decrease by a comparable amount, indicating that bottom depth is a key factor in setting their speed. In fact, the nearness of the speeds in the deep basin to cpM2, together with their twice-daily generation, suggest that they begin as a semidiurnal internal tide. However, A waves (red), which are larger, travel faster than B waves (blue) in the deep basin, indicating that nonlinearity also plays a role. The deep-basin speeds are generally consistent with that measured for an energetic wave near 119.25°E by Klymak et al. (2006; asterisk).
The ratio of observed speed to the linear irrotational speed cp/ce (Fig. 5e) begins at about 1.18 and 1.12 times ce in the deep basin for A and B waves, respectively (1.11 and 1.05 times cpM2). Generally, cp/ce increases moving westward, reaching 1.4 and 1.6 for A and B waves between LR3 and MP2, respectively. At the foot of the continental slope, A waves propagate at 1.38ce.
b. Modulation of NLIW speed by the diurnal internal tide
The differences between A and B waves are explored in more detail in Fig. 6, which plots speed and amplitude at two representative mooring pairs, N1–MP1 (right) and LR1–LR2 (left). In the deep basin approaching the continental slope (right), observed speed (top) shows a clear alternation between the faster A waves and their slower B counterparts. Given that the A waves are larger (bottom), their greater speed is attributed to the effects of nonlinearity. Lacking a theoretical relation between amplitude and speed in a rotating frame, observed speeds are compared to KdV predictions [Eq. (1)] using the measured amplitude of the waves (dashed). KdV underpredicts both the increase above the linear speed and the magnitude of the difference between the large and small waves. The excess is greater than our error estimates (appendix A), apparently indicating that KdV is not an adequate description of the waves.
On the upper continental slope, B waves clearly travel faster (left, top) in spite of their smaller amplitude (left, bottom). We associate this unexpected finding with modulation of the speed of every other wave by the diurnal internal tide. This is made clear by examination of the relationship of the nonlinear waves with the internal tide at A1, A2, MP1, and LR1 (Fig. 7). Displacement anomaly at A1 and A2 (top two panels) and zonal current averaged between 100 and 200 m at MP1 and LR1 (bottom two panels) are plotted in black and decomposed into their semidiurnal (thin gray) and diurnal (heavy gray) components via bandpass filtering. Comparing with Figs. 3 and 4, it is clear that the low-frequency velocity signals are proxies for the baroclinic internal tide rather than the barotropic tide.
Waves are mostly absent at A1, appearing at A2 as sharp spikes superimposed on the internal tide troughs, which are dominated by the semidiurnal components (light gray). Moving westward to MP1, a diurnal inequality develops as diurnal signals increase, but the nonlinear waves still descend from each internal tide trough (though advanced slightly, as observed by ZA06). However, at LR1 the character of the internal tide changes dramatically, becoming dominated by the diurnal component. Examination of the depth–time plots (Fig. 4a) shows that this is the upper portion of a low-mode diurnal internal tide, likely locally generated as argued by Duda et al. (2004) and Duda and Rainville (2008). Now, all waves still show westward shallow flow (downward spikes), but the odd-numbered A waves now oppose the low-frequency baroclinic flow, whereas the B waves are aligned with it.
The effect of this diurnal internal tide on the twice-daily nonlinear waves is to speed up the B waves and slow down the A waves through the effects of background shear, as demonstrated by solving the Taylor–Goldstein equation [Eq. (A3)] using the observed background velocity profiles of the diurnal tide (Fig. 8). For a westbound wave propagating past LR1 in zero background flow (gray), the eigenspeed is 1.5 m s−1. Using the velocity profile 1 h prior to wave 11 (thick line), the baroclinic currents of the diurnal internal tide slow the wave down by 15%. When the flow reverses for the next wave (thin line), the eigenspeed is increased 15%. The resulting speedup–slowdown (Fig. 6b, left, gray line) is of the correct magnitude and sign to explain the observed speed differences and to overcome the effects of KdV nonlinearity (dashed).
Vertical motion of the internal tide can also modulate ce by displacing the thermocline. A westward-propagating internal tide would have upward displacement during shallow eastward flow. Hence, the effect would reinforce that of the baroclinic currents. Though internal tide displacements were not directly measured on the upper slope, we calculate that upward vertical displacements of 50 and 80 m would slow the waves by 10% and 16%, respectively. Accurate modeling of the effects of the background internal tide on the NLIW will require consideration of these effects together, as well as nonhydrostatic effects, likely in a fully nonlinear model.
5. Structure and evolution
a. Width, amplitude, and energy
Returning to Fig. 2b, we see qualitatively the progression of the waves from broad, near-sinusoidal waveforms in the deep basin to narrow trains of waves approaching and transiting the upper continental slope. Structural changes undergone by the waves transiting the basin are next examined more quantitatively (Fig. 9). Though not computed at A1, widths there appear >80 km (Fig. 2b). They begin narrowing in the deep basin, having already reduced their width to 4–15 km at A2. All waves continue to show a clear decrease in width as they enter shallower water, reaching 150–800 m at LR3. A waves (red) show a clear tendency to be narrower at all locations. Wave amplitude (Fig. 9b) is not computed at the PIES moorings but increases sharply from MP1 to LR1 as the waves transit the upper continental slope, only to decrease again at LR3. A waves are approximately a factor of 2 larger than B waves at all locations.
In spite of the increased amplitude on the upper slope, the smaller width causes kinetic energy (Fig. 2b) to remain approximately constant from N1 to LR1 before dropping markedly by a factor of ∼2 from LR1 to LR3. Hence, assuming equipartition of kinetic and potential energy (Moum et al. 2007), the waves approximately conserve energy until near LR2, where energy decreases because of 1) development of trailing waves and 2) dissipation resulting from a variety of processes including trapped cores. The waves’ energy budget on the upper slope is discussed in more detail by R.-C. Lien et al. (2010, unpublished manuscript).
b. Wave trains
Returning to Fig. 5a, the development of trailing waves is examined next (gray numbers). No multiple waves are observed east of MP1; at MP1, all A waves besides the first two have developed one or two trailing waves. All B waves remain single troughs until LR1, but most have developed trailing waves by LR2. A waves spawn as many as eight waves; B waves all develop fewer than two trailing waves. Hence, the number of trailing waves depends on amplitude and/or width, consistent with the larger, more nonlinear waves being more unstable to the development of wave trains.
We detected 14 waves at sites spanning the SCS basin and tracked their speed, wavelength, number of crests, and energy over 560 km of their transit. We find the following:
Observed trajectories are consistent with propagation across the deep basin at speeds of 1.18 and 1.12 times ce in the deep basin for A and B waves, respectively (1.11 and 1.05 times cpM2). Attempts to ascertain the phase of generation were inconclusive, emphasizing the need for more detailed observations near the ridges while monitoring the times of the wave arrivals in the west.
Waves in the deep basin begin as nearly sinusoidal, semidiurnal waveforms with widths of many tens of kilometers and speeds in slight excess of cpM2, suggesting they begin as weak nonlinearities in the semidiurnal internal tide before developing into NLIW near the continental slope. There, their overall speed decreases, but the excess above linear speed increases, until the waves travel ∼1.5ce (∼1.4cpM2) at the western end of the array. The observed speeds are faster than predicted by KdV theory.
On the upper continental slope, B waves travel faster than A waves, in spite of their smaller amplitude. We attribute this to modulation of the NLIW propagation speed by a diurnal internal tide on the upper slope, where the daily reversing baroclinic current introduces “background” shear for the NLIW, alternately speeding/slowing the linear speed of the twice-daily waves.
All detected waves are mode 1 and are initially solitary until MP1, where trailing waves begin to develop. Their width decreases moving westward from ∼80 km to ∼200 m. Their amplitude increases sharply as the waves move onto the continental slope; however, the corresponding decrease in width leads to approximate conservation of kinetic energy until about LR2. KE drops at LR3, presumably because of development of trailing waves and dissipative processes, as discussed in R.-C. Lien et al. (2010, unpublished manuscript).
This study has used data across the SCS basin to document speed and structure changes in a single group of 14 waves from near their generation site to the middle of the upper continental slope. The waves’ initial appearance at A1 as broad, nearly sinusoidal pulses and steepening by A2 into broad but identifiable NLIW is consistent with a lack of observed wave crests east of 120.5°E (Zhao et al. 2004; J09). Both of these observations support the notion that the NLIW emerge from the generation region as internal tides, becoming unstable in the deep basin [possibly strongly affected by rotational dispersion, as suggested by Helfrich and Grimshaw (2008) and Farmer et al. (2009)]. The need to understand this process, along with the inconsistency of our observed speeds with KdV predictions, underscores the need for a better description of rotation-affected NLIW.
For these data, the broad extent of the array came at a cost in the overall overlapping time series length. We therefore wish to emphasize that the results presented are only valid for this particular 10-day period. For example, preliminary examination of year-long records from the LR1, LR2, and LR3 moorings indicates that our observation of smaller B waves traveling faster on the upper slope than the larger A waves is true during much of the year, but not at all times. This is not surprising, because the effect would be expected to depend on the phasing between the mixed tides at Luzon Strait that generate the NLIW and the locally generated diurnal tide on the upper slope that modulates their speed. In addition, low-frequency modulation of stratification and currents should affect both the generation of the waves, as well as their propagation speeds (ZA06). In emphasis of this expected but poorly understood amplitude modulation, all of the waves in this study are wider, smaller, and less energetic than the deep-basin soliton observed by Klymak et al. (2006). Klymak’s observations (asterisks) appear to have occurred during a more energetic time. Detailed reports on the seasonality of NLIW in year-long records will be reported for the deep basin and upper slope by Ramp et al. (2010, unpublished manuscript) and Lien et al. (2010, unpublished manuscript). Placing large waves such as that seen by Klymak et al. (2006) into a seasonal context should be a high priority.
The difficulties in determining the phase of generation are understandable because of the past lack of data in the deep basin. The current study has shown that speeds there are in excess of linear speeds by O(10%). A 10% speed difference integrated across the basin results in 6–8-h differences in travel time, or >180° uncertainty in semidiurnal phase. Even in the present study, the complexity of the near-field signals in the 3D, two-ridge system, the difficulty of placing moorings close to the sills (resulting in lingering uncertainty regarding the barotropic currents estimated from the N2 data) and the large extent of the generation region present real challenges. In spite of these, it is hoped that these results will catalyze modeling and observational efforts needed for progress.
This work was supported by the Office of Naval Research under Grant N00014-05-1-0283. We thank David Farmer for use of his PIES data. We are indebted to Andrew Cookson; Eric Boget; Marla Stone; Fred Bahr; and the Taiwanese mooring technicians, particularly Wen-Hwa Her, for their expertise and skill in design, deployment, and recovery of the moorings and to the steady hands of the captains and crews of R/V OR1 and OR3 (Taiwan) and Melville (United States). Finally, we wish to thank two anonymous reviewers for their efforts and helpful comments.
Theoretical Wave Speeds
Under the hydrostatic approximation, the vertical velocity W(z) of a mode-1, periodic traveling wave is governed by the Sturm–Liouville equation (Gill 1982),
where N(z) is the stratification and ce is the eigenspeed. Without rotation (inertial frequency f = 0), long linear internal waves are nondispersive and the eigenspeed equals the phase and group speeds: ce = cp = cg. Rotation does not affect the shape of the eigenmodes or the value of ce, but the resulting dispersion causes the group ∂ω/∂k and phase speed ω/k to differ from the eigenspeed ce (Alford et al. 2006). Hence, ce has no physical meaning for waves with rotation but is related to the phase and group speeds via the dispersion relation. In the SCS, waves of frequency M2 and K1 travel at phase speeds cpM2 and cpK1 of 1.07 and 1.5 times ce, respectively.
Relaxing the hydrostatic assumption, (A1) becomes
where ω is the wave frequency. The nonhydrostatic eigenspeed ĉe depends on frequency (or wavelength via the dispersion relation) but is always less than the hydrostatic speed.
b. Calculation and verification
Linear eigenspeed ce is computed by solving (A1) using a shooting method. Several solvers were used, with all agreeing to within 5%. The calculation was performed on density profiles at ⅙° intervals from the Generalized Digital Environmental Model (GDEM) database climatology for the month of April (Teague et al. 1990) and bathymetry from the Naval Research Laboratory (NRL).
Spatial and temporal variability not captured in the climatology introduce errors. These are evaluated by comparing ce computed from climatology and from measured profiles (examples in Fig. A1) from three hydrographic lines conducted in April 2005 and April/July 2007 (asterisks in Fig. 1), and the MP1 profiler. All casts were done from the surface to within 20 m of the bottom, with the exception of MP1, where no data were collected below 80 m. There, repeated nearby CTD casts conducted by R. Pinkel and J. Klymak (2007, personal communication) were substituted above 80 m.
Stratification differences across Luzon Strait are sufficient to have a major impact on wave speeds. Profiles measured in the SCS (green and black) agree well with each other and climatology (blue) but show a much shallower thermocline than the Pacific side (red). Associated eigenspeeds, computed assuming H = 2000 m to isolate effects of stratification, are 2.35–2.42 m s−1 in the SCS but 2.88 m s−1 in the Pacific.
Eigenspeeds computed using measured density from the time of the cruise (Fig. A2a, green) are in excellent agreement with the April climatology (blue) and with the time mean value at MP1. Values computed using the 2005 and 2007 April values and even the July values (dashed) differ little. The largest differences are between longitudes 118° and 119°E, where MP1 and transect speeds exceed those from the climatology by about 3%. In spite of this, climatological values are used for simplicity.
To emphasize the importance of the spatial stratification differences near Luzon Strait, transects of speed were computed using NRL bathymetry but substituting horizontally constant profiles representative of the Pacific (red) and the SCS (black). As shown earlier, using the Pacific profile results in speeds about 20% higher than should be relevant for propagation in the SCS. Likewise, the basin values (black) are too low near the ridge system. The values from the transect (green) and climatology (blue) bridge these two profiles.
Temporal variability is assessed next using the time series of density profiles taken at MP1. Time averages are first taken in 3-day moving windows to minimize tidal isopycnal displacements. Stratification (Fig. A3a) varies only little except for a brief period near yearday 140. Associated eigenspeed (Fig. A3b), again computed at 2000 m for comparison with Fig. A1, bracket the value from the April 2007 CTD taken nearby (green) and are in slight excess (∼2%) of the climatology (blue). Variations are ∼0.04 m s−1 (3%). A slight trend is perhaps seen toward the end of the record, but overall a temporally constant value seems justified. Taking all of these effects together, we estimate errors of O(4%).
Errors in Observed Speed
Errors in speed are all traceable to errors in either arrival time and/or location. Both of these increase as mooring spacing decreases. At the closest pair, upper-bound clock drift errors of ∼120 s, measured by comparison of GPS and instrument time before and after recovery, result in 1.4% error on the 2.4-h mean travel time. In like manner, upper-bound uncertainties in mooring position and lateral spread of beams introduce position uncertainties of approximately 100 and 200 m, respectively. These result in 0.6% and 1.2% errors on the 15-km separation distance. The worst-case total fractional error is then given by the sum of all three, 3.2%.
Greater uncertainty results from the choice of the trajectory along which speed is computed and the method used in determining mooring positions along it (Fig. B1). To evaluate these, mean speed over all waves was computed using several different trajectories; the one shown in Fig. 1 (black) and a zonal one along 21°N (gray) are plotted. In addition, an alternate projection method, termed “nearest,” was computed as the intersection between the track and a perpendicular passing through the mooring. This method would be appropriate if no radial spreading occurred (straight wave crests).
Speeds from pairs west of 118°E are quite insensitive to either the method or trajectory used. Closer to the source, the zonal trajectory gives values 8% slower at the far east and 9% faster for the cylindrical method at N1–MP1. The greatest uncertainty arises at N1–MP1 because of the closeness of the pair and the large off-track distance of N1. There, the fastest value (nearest projection along the 21°N track) is 12% faster than the cylindrical value at the main track. Thus, 12% may be used as a conservative upper bound on the errors associated with track geometry at these pairs; 3%–4% is more appropriate for the others.
Corresponding author address: Matthew H. Alford, 1013 NE 40th St., Seattle, WA 98105. Email: firstname.lastname@example.org